Counting Radial Nodes between Orbitals with Same l

[sams]

Dear Everyone,

Could anyone explain why we count only the number of radial nodes between the subshells that have the same orbital angular momentum l ?

For example, 3p-orbitals have 1 radial node that exists between the 3p- and 2p-orbitals.
Shouldn't be there additional radial nodes that exist between the 3p- and 2s-orbitals and between the 3p- and 1s-orbitals?

Thank you so much for your help...

 

[DrClaude]

When the radial equation is solved, it is for a given value of $l$. Solutions are found under the condition that $n > l$. Hence, for $l=1$, the first radial wave function is 2p, and thus has 0 node; the second wf is 3p, and has 1 node, etc.

For $l=0$, the first radial wave function, 1s, has no nodes, just like the 2p, so you can't use the number of nodes to compare the wave functions for different $l$.

 

[sams]

you can't use the number of nodes to compare the wave functions for different $l$.

Thank you so much for your detailed reply. But what is the reason behind this comparison? In other words, is there any physical meaning that cause the radial nodes to be created only for same orbital angular momentum l subshells?

 

[DrClaude]

Thank you so much for your detailed reply. But what is the reason behind this comparison? In other words, is there any physical meaning that cause the radial nodes to be created only for same orbital angular momentum l subshells?

I said it already: because the radial equation
$$
- \frac{\hbar^2}{2 \mu} \nabla^2 R(r) + \left[ - \frac{Z e^2}{2 \pi \epsilon_0 r} + \frac{l (l+1) \hbar^2}{2 \mu r^2} \right] R(r) = R(r)
$$
is solved for a given value of $l$. You get a series of solutions $R_{n,l}$ with $n>l$ for each value of $l$, and it is those solutions that display the usual increasing number of nodes.